Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
456 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D83 |
| commutator | C22×C83 | C23×C4 |
| Fitting | #384×C83 | C2 |
| socle, cosocle | C23×C83 | C24 |
| prime radical | C23×D83 | C23 |
| hypercenter | #384 | D83 |
| hypocenter | C83 | #2383 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #19127596 | C2 | semidirect |
| 4. #19127582 | C2 | semidirect |
| 6. #19127569 | C2 | semidirect |
| 7. #384×C83 | C2 | semidirect |
| 9. #14581786 | C4 | semidirect |
| 28. Dic83⋊C4 | C2×C4 | semidirect |
| 37. C42×C83 | C2×C4 | semidirect |
| 73. C83 | #2383 | semidirect |
| 79. #384 | D83 | semidirect |
| 110. C42 | C4×D83 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#14581786 | 2 |
| #19127596 | 2 |
| #19127591 | 2 |
| #19127582 | 2 |
| #19127572 | 2 |
| #19127569 | 2 |
| #384×C83 | 2 |
| #2383 | 83 |